820 research outputs found
An implementation of Sub-CAD in Maple
Cylindrical algebraic decomposition (CAD) is an important tool for the
investigation of semi-algebraic sets, with applications in algebraic geometry
and beyond. We have previously reported on an implementation of CAD in Maple
which offers the original projection and lifting algorithm of Collins along
with subsequent improvements.
Here we report on new functionality: specifically the ability to build
cylindrical algebraic sub-decompositions (sub-CADs) where only certain cells
are returned. We have implemented algorithms to return cells of a prescribed
dimensions or higher (layered {\scad}s), and an algorithm to return only those
cells on which given polynomials are zero (variety {\scad}s). These offer
substantial savings in output size and computation time.
The code described and an introductory Maple worksheet / pdf demonstrating
the full functionality of the package are freely available online at
http://opus.bath.ac.uk/43911/.Comment: 9 page
Theorem of three circles in Coq
The theorem of three circles in real algebraic geometry guarantees the
termination and correctness of an algorithm of isolating real roots of a
univariate polynomial. The main idea of its proof is to consider polynomials
whose roots belong to a certain area of the complex plane delimited by straight
lines. After applying a transformation involving inversion this area is mapped
to an area delimited by circles. We provide a formalisation of this rather
geometric proof in Ssreflect, an extension of the proof assistant Coq,
providing versatile algebraic tools. They allow us to formalise the proof from
an algebraic point of view.Comment: 27 pages, 5 figure
On mechanical quantifier elimination for elementary algebra and geometry
We give solutions to two problems of elementary algebra and geometry: (1) find conditlons on real numbers p, q, and r; so that the polynomial function f(x) = x4 + px2 + q x+ r is nonnegative for all real x and (2) find conditions on real numbers a, b, and c so that the ellipse (x−c)2q2+y2b2−1=0 lies inside the unit circle y2 + x2 - 1 = O. Our solutions are obtained by following the basic outline of the method of quantifier elimination by cylindrical algebraic decomposition (Collins, 1975), but we have developed, and have been considerably aided by, modified vcrsions of certain of its steps. We have found three equally simple but not obviously equivalent solutions for the first problem, illustrating the difficulty of obtaining unique “simplest” solutions to quantifier eliminetion problems of elementary algebra and geometry
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Towards justifying computer algebra algorithms in Isabelle/HOL
As verification efforts using interactive theorem proving grow, we are in need of certified algorithms in computer algebra to tackle problems over the real numbers. This is important because uncertified procedures can drastically increase the size of the trust base and under- mine the overall confidence established by interactive theorem provers, which usually rely on a small kernel to ensure the soundness of derived results.
This thesis describes an ongoing effort using the Isabelle theorem prover to certify the cylindrical algebraic decomposition (CAD) algorithm, which has been widely implemented to solve non-linear problems in various engineering and mathematical fields. Because of the sophistication of this algorithm, people are in doubt of the correctness of its implementation when deploying it to safety-critical verification projects, and such doubts motivate this thesis.
In particular, this thesis proposes a library of real algebraic numbers, whose distinguishing features include a modular architecture and a sign determination algorithm requiring only rational arithmetic. With this library, an Isabelle tactic based on univariate CAD has been built in a certificate-based way: external, untrusted code delivers solutions in the form of certificates that are checked within Isabelle. To lay the foundation for the multivariate case, I have formalised various analytical results including Cauchy’s residue theorem and the bivariate case of the projection theorem of CAD. During this process, I have also built a tactic to evaluate winding numbers through Cauchy indices and verified procedures to count complex roots in some domains.
The formalisation effort in this thesis can be considered as the first step towards a certified computer algebra system inside a theorem prover, so that various engineering projections and mathematical calculations can be carried out in a high-confidence framework
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